homework and exercises - Projectile/orbital motion over very long distance

We know the optimum angle for greatest horizontal displacement when launching an object with projectile motion is 45 degrees. How to solve the angle when it is real long distance around the earth where we cannot assume that gravity is constant but is given by $F=\frac {-GMm}{r^2}$. I use differetial equation to solve it and obtain 45 degrees.

First, for the vertical distance $y$, change $v=u-gt$ to $y'=u-y''t$. Then solve it to obtain a function $y(t)$. When $y'(t)=0$ we have the time for maximum vertical distance. Then insert the value to horizontal equation.

Can anyone guide me through this problem because I am not confident with my working